Adding Integers
Learning Objective(s)
· Add two or more integers with the same sign.
· Add two or more integers with different signs.
Introduction
On an extremely cold day, the temperature may be −10. If the temperature rises 8 degrees, how will you find the new temperature? Knowing how to add integers is important here and in much of algebra.
Since positive integers are the same as natural numbers, adding two positive integers is the same as adding two natural numbers.
Try out the interactive number line below. Choose a few pairs of positive integers to add. Click and drag the blue and red dots, and watch how the addition works.
To add integers on the number line, you move forward, and you face right (the positive direction) when you add a positive number.
As with positive numbers, to add negative integers on the number line, you move forward, but you face left (the negative direction) when you add a negative number.
In both cases, the total number of units moved is the total distance moved. Since the distance of a number from 0 is the absolute value of that number, then the absolute value of the sum of the integers is the sum of the absolute values of the addends.
When both numbers are negative, you move left in a negative direction, and the sum is negative. When both numbers are positive, you move right in a positive direction, and the sum is positive.
| To add two numbers with the same sign (both positive or both negative):
· Add their absolute values and give the sum the same sign. |
| Example | |
| Problem | Find −23 + (−16). |
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Both addends have the same sign (negative). |
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| So, add their absolute values: |−23| = 23 and |−16| = 16.
The sum of those numbers is 23 + 16 = 39. |
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Since both addends are negative, the sum is negative. |
| Answer | −23 + (−16) = −39 |
With more than two addends that have the same sign, use the same process with all addends.
| Example | |
| Problem | Find −27 + (−138) + (−55). |
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All addends have the same sign (negative). |
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| So, add their absolute values: |−27| = 27, |−138| = 138, and |−55| = 55.
The sum of those numbers is 27 + 138 + 55 = 220. |
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| Since all addends are negative, the sum is negative. |
| Answer | −27 + (−138) + (−55) = −220 |
| Find −32 + (−14).
A) 46 B) 18 C) −18 D) −46
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What happens when the addends have different signs, like in the temperature problem in the introduction? If it’s −10 degrees, and then the temperature rises 8 degrees, the new temperature is −10 + 8. How can you calculate the new temperature?
Using the number line below, you move forward to add, just as before. Face and move in a positive direction (right) to add a positive number, and move forward in a negative direction (left) to add a negative number.
Try adding integers with different signs with this interactive number line. See if you can find a rule for adding numbers without using the number line.
Notice that when you add a positive integer and a negative integer, you move forward in the positive (right) direction to the first number, and then move forward in the negative (left) direction to add the negative integer.
Since the distances overlap, the absolute value of the sum is the difference of their distances. So to add a positive number and a negative number, you subtract their absolute values (their distances from 0.)
What is the sign of the sum? It’s pretty easy to figure out. If you moved further to the right than you did to the left, you ended to the right of 0, and the answer is positive; and if you move further to the left, the answer is negative.
If you didn’t have the number line to refer to, you can find the sum of −1 + 4 by
· subtracting the distances from zero (the absolute values) 4 – 1 = 3 and then
· applying the sign of the one furthest from zero (the largest absolute value). In this case, 4 is further from 0 than −1, so the answer is positive: −1 + 4 = 3
Look at the illustration below.
If you didn’t have the number line to refer to, you can find the sum of −3 + 2 by
- subtracting the distances from zero (the absolute values) 3 – 2 = 1 and then
- applying the sign of the one furthest from zero (the largest absolute value). In this case, |−3| > |2|, so the answer is negative: −3 + 2 = −1
| To add two numbers with different signs (one positive and one negative):
· Find the difference of their absolute values. · Give the sum the same sign as the number with the greater absolute value. |
Note that when you find the difference of the absolute values, you always subtract the lesser absolute value from the greater one. The example below shows you how to solve the temperature question that you considered earlier.
| Example | |
| Problem | Find 8 + (−10). |
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The addends have different signs. So find the difference of their absolute values. |
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| |−10| = 10 and |8| = 8.
The difference of the absolute values is 10 – 8 = 2. |
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| Since 10 > 8, the sum has the same sign as −10. |
| Answer | 8 + (−10) = −2 |
| Example | ||
| Problem | Evaluate x + 37 when x = −22. | |
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| x + 37 −22 + 37 | Substitute −22 for x in the expression. |
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| |−22| = 22 and |37| = 37 37 – 22 = 15
| The addends have different signs. So find the difference of their absolute values.
Since |37| > |−22|, the sum has the same sign as 37. |
| Answer | −22 + 37 = 15 |
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With more than two addends, you can add the first two, then the next one, and so on.
| Example | ||
| Problem | Find −27 + (−138) + 55. | |
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| Add two at a time, starting with −27 + (−138).
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| |−27| = 27 and |−138| = 138 27 + 138 = 165 −27 + −138 = −165
| Since they have the same signs, you add their absolute values and use the same sign. |
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| −165 + 55
|−165| = 165 and |55| = 55 165 – 55 = 110
−165 + 55 = −110
| Now add −165 + 55. Since −165 and 55 have different signs, you add them by subtracting their absolute values.
Since 165 > 55, the sign of the final sum is the same as the sign of −165. |
| Answer | −27 + (−138) + 55 = −110 |
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| Find 32 + (−14).
A) 46 B) 18 C) −18 D) −46
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Summary
There are two cases to consider when adding integers. When the signs are the same, you add the absolute values of the addends and use the same sign. When the signs are different, you find the difference of the absolute values and use the same sign as the addend with the greater absolute value.
